<p><span style="color:#27ae60;"><span style="font-size:20px;">Solving the Problem</span></span></p>
<p>Let's solve this <strong><span style="color:#8e44ad;">Piece by Piece</span></strong>:</p>
<blockquote>
<p>If $r\le s\le t\le u\le v\le 110$</p>
</blockquote>
<p>This inequality tells us that $r,s,t,u$ and $v$ can each be less than or equal to $110$.</p>
<blockquote>
<p>and the average (arithmetic mean) of r, s, t, u, and v is 100,</p>
</blockquote>
<p>From this, we can deduce the sum of $r,s,t,u$ and $v$ :</p>
<ul>
<li>$sum = mean * \#\,values$</li>
<li>$sum = 100 * 5$</li>
<li>$sum = 500$</li>
</ul>
<blockquote>
<p>what is the least possible value of r?</p>
</blockquote>
<p>Now, we should realize that the question is telling us to <strong>minimize</strong> the value of $r$. Since the mean is fixed at $100$ and therefore the sum is also fixed at $500$, in order to reduce the value of $r$, we would need to <strong>maximize</strong> the values of $s,t,u$, and $v$:</p>
<ul>
<li>We know that $s,t,u$, and $v$ can be as big as $110$ from our inequality</li>
<li>So, we can set $s,t,u$, and $v$ to $110$</li>
</ul>
<p>Now, we just need to figure out what the value of $r$ should be to get our desired total sum:</p>
<ul>
<li>$total\,sum = r + sum\,of\,s,t,u,\,and\,v$</li>
<li>$sum\,of\,s,t,u,\,and\,v = 110 * 4 = 440$</li>
<li>$500 = r + 440$</li>
<li>$60 = r$</li>
</ul>
<p>So, the correct answer would be <span style="color:#27ae60;">D</span>.</p>
<p> </p>